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I'm new to topology, and I came across this definition on topology:

Given a topological space (X, T), a topology is defined to be a collection of open subsets which satisfies the following properties: 1. empty and X are in T 2. union of arbitrary collection of sets in T are open 3. intersection of a finite number of sets in T are open.

I'm quite confused if the only sets to be considered open are the ones in T, or some of the other subsets not in T can also be considered open. Can anyone give me light to this?

Also, are all sets in T are considered to be open and closed at the same time?

Thanks!

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  • $\begingroup$ Yes, $T$ is the set of all open subsets of $X$. $\endgroup$ – saulspatz Feb 17 '18 at 7:58
  • $\begingroup$ Thanks! And insults are humbly taken :) $\endgroup$ – Jonelle Yu Feb 17 '18 at 8:05
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    $\begingroup$ @user8734617 . Someone totally new to topology might read the def'n of a topology and elsewhere see the term "open set" and not realize it means a member of the topology. $\endgroup$ – DanielWainfleet Feb 17 '18 at 9:22
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    $\begingroup$ @JonelleYu I've deleted my comment. It was not my intention that it sounded insulting, very sorry if I came across in such way. I only meant we needed to find out what exactly makes you feel uneasy with this notion of "open set as a member of $T$" - it comes across that you possibly have some thoughts which make you believe that sometimes sets outside $T$ should be considered open, so I wanted to know more about that, to make it easier to argue one way or another. I am sorry again, comments are restricted in length, I tried to be terse and came across as unpleasant. Apologies! $\endgroup$ – user491874 Feb 17 '18 at 9:34
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If $(X,\tau)$ is a topological space, then the open subsets of $X$ are the elements of $\tau$ and only the elements of $\tau$.

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  • $\begingroup$ Thanks! Also I've read that (X,T) can be defined in terms of closed sets. Does this imply that the sets in T are open and closed at the same time? Hence the only sets to be considered open and closed sets of X are those in T? $\endgroup$ – Jonelle Yu Feb 17 '18 at 8:02
  • $\begingroup$ Not at all. It means that we define a topology on a set $X$ by giving the set $\mathcal F$ of closed subsets of $X$. Then the open subsets of $X$ will be the sets of the type $X\setminus F$, with $F\in\mathcal F$. $\endgroup$ – José Carlos Santos Feb 17 '18 at 8:05
  • $\begingroup$ Okay, I get it, thanks a lot! $\endgroup$ – Jonelle Yu Feb 17 '18 at 8:06
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It is wrong and confusing if a topology $\tau$ on set $X$ is defined as a collection of open subsets of $X$ such that....

The word "open" must be left out since at time of defining you actually do not know yet what it means.

This misstep might have been the source of your confusion.

A topology $\tau$ on set $X$ should be defined as a collection of subsets of $X$ such that...

Afterwards the concept "open subset" can be defined. If set $X$ is equipped with a topology $\tau$ then a subset $A$ of $X$ is open if $A\in\tau$.

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